WASHINGTON — Outlines of a possible compromise that would more quickly deport minors arriving from Central America emerged Thursday as part of President Barack Obama’s $3.7 billion emergency request to address the immigration crisis on the nation’s southern border.

Republicans demanded speedier deportations, which the White House initially had supported but left out of its proposal after complaints from immigrant advocates and some Democrats. On Thursday, the top House and Senate Democrats pointedly left the door open to them.

“It’s not a deal-breaker,” said House Minority Leader Nancy Pelosi, D-Calif. “Let them have their face-saver. But let us have the resources to do what we have to do.” Her spokesman Drew Hammill later clarified that any changes “must ensure due process for these children.”

In the Senate, Majority Leader Harry Reid, D-Nev., said: “I’m not going to block anything. Let’s see what comes to the floor.”

But opposition arose late in the day from key Democratic senators, suggesting battles ahead before any deal could be struck.

“I can assure you that I will fight tooth and nail changes in the Trafficking Victims Protection Act,” Senate Judiciary Chairman Patrick Leahy said at a hearing on the situation, referring to the law Republicans want to change.

Noting that the arriving migrants include young girls trying to escape sex violence and gangs, Leahy said: “I’m not sure Americans all really feel we should immediately send them back.”

Reid and Pelosi made their comments as House Speaker John Boehner, R-Ohio, and Senate Republican leader Mitch McConnell, R-Ky., both said they didn’t want to give Obama a “blank check” to deal with the crisis of tens of thousands of unaccompanied children arriving at the Texas border, many fleeing gangs and drawn by rumors they would be able to stay in the U.S. Boehner and McConnell indicated policy changes would be necessary to win their support.

“We want to make sure we actually get the right tools to help fix the problem,” McConnell said. Obama “needs to work with us to get the right policy into effect.”

Proponents of speedier deportations say an effective way to stem the tide of young immigrants crossing the border would be to send them back home right away, to show their parents that the trip north was wasted.

The developments came as Obama’s Homeland Security secretary, Jeh Johnson, defended the emergency spending request at a hearing of the Senate Appropriations Committee. He said that without the money, the U.S. Immigration and Customs Enforcement and Customs and Border Patrol agencies would both run out of money in the next two months, and the Homeland Security Department “would need to divert significant funds from other critical programs just to maintain operations.”

At issue is a law approved in 2008. Passed to give protection to sex trafficking victims, it requires court hearings for migrant young people who arrive in this country from “noncontiguous” countries — anywhere other than Mexico or Canada.

Because of enormous backlogs in the immigration court system, the result in the current crisis is that kids streaming in from El Salvador, Honduras and Guatemala are released to relatives or others in the U.S. with notices to appear at long-distant court hearings that many of them never will attend.

Republicans want the government to have the authority to treat Central American kids the same way as kids from Mexico, who can be removed quickly unless they convince Border Patrol that they have a fear of return that merits additional screening.

“I think clearly we would probably want the language similar to what we have with Mexico,” Boehner said.

White House officials have said they support such changes and indicated last week that they would be offering them along with the emergency spending request. But immigration advocates objected strongly, saying children would be denied legal protections, and the White House has not yet made a formal proposal.

Asked Thursday about the issue, Johnson said he supported changing the law to treat children from Central American nations the same as those from Mexico.

“We want the flexibility in the current situation to have that discretion,” he said.

But in response to concerns voiced by Leahy and Sens. Dick Durbin, D-Ill., and Tom Harkin, D-Iowa, Johnson insisted that the kids still would be protected.

“A request for discretion, as long as I’m secretary, means a request for the ability to do the right thing,” he said.

The comment didn’t quiet Democrats’ concerns.

“You want flexibility. There’s danger in flexibility,” Harkin said. “The single most important thing is to take care of these kids.”

Advocates said they remained strongly opposed to such policy changes and expressed anger that after comprehensive immigration reform failed to advance in Congress this year, lawmakers may be headed toward a vote on deporting minors more quickly.

“They weren’t able to get immigration reform done in this Congress and this is going to be the only piece of immigration that gets done, a bill that says we’re going to deport children fleeing violence faster,” said Marielena Hincapie, executive director of the National Immigration Law Center. “If Democrats can’t stand up to this and be the party that’s protecting children and refugees, it’s a really sad day for the country.”

More than 57,000 unaccompanied children have arrived since October even as tens of thousands more have arrived traveling as families, mostly mothers with their children.

Many are trying to reunite with family members and to escape a spike in violence in their countries, but they also report hearing rumors that once here, they would be allowed to stay. Republicans blame Obama policies aimed at curbing deportations of immigrants brought into the country illegally as children for contributing to those rumors, something Obama administration officials have largely rejected.

The situation has complicated the already rancorous debate over remaking the nation’s immigration laws at a moment when Obama has declared legislative efforts to do so dead and announced plans to proceed on his own executive authority to change the flawed system where he can.